Solve 3100=left(x-10right)left(920-2xright)

$$3100 = \left( x-10 \right) \left( 920-2x \right)$$

Solve for x

$$x=5\sqrt{1963}+235\approx 456.528779169$$

$$x=235-5\sqrt{1963}\approx 13.471220831$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-10 by 920-2x and combine like terms.

$$3100=940x-2x^{2}-9200$$

Swap sides so that all variable terms are on the left hand side.

$$940x-2x^{2}-9200=3100$$

Subtract 3100 from both sides.

$$940x-2x^{2}-9200-3100=0$$

Subtract 3100 from -9200 to get -12300.

$$940x-2x^{2}-12300=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-2x^{2}+940x-12300=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 940 for b, and -12300 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-940±\sqrt{940^{2}-4\left(-2\right)\left(-12300\right)}}{2\left(-2\right)}$$

Square 940.

$$x=\frac{-940±\sqrt{883600-4\left(-2\right)\left(-12300\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-940±\sqrt{883600+8\left(-12300\right)}}{2\left(-2\right)}$$

Multiply 8 times -12300.

$$x=\frac{-940±\sqrt{883600-98400}}{2\left(-2\right)}$$

Add 883600 to -98400.

$$x=\frac{-940±\sqrt{785200}}{2\left(-2\right)}$$

Take the square root of 785200.

$$x=\frac{-940±20\sqrt{1963}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-940±20\sqrt{1963}}{-4}$$

Now solve the equation x=\frac{-940±20\sqrt{1963}}{-4} when ± is plus. Add -940 to 20\sqrt{1963}.

$$x=\frac{20\sqrt{1963}-940}{-4}$$

Divide -940+20\sqrt{1963} by -4.

$$x=235-5\sqrt{1963}$$

Now solve the equation x=\frac{-940±20\sqrt{1963}}{-4} when ± is minus. Subtract 20\sqrt{1963} from -940.

$$x=\frac{-20\sqrt{1963}-940}{-4}$$

Divide -940-20\sqrt{1963} by -4.

$$x=5\sqrt{1963}+235$$

The equation is now solved.

$$x=235-5\sqrt{1963} x=5\sqrt{1963}+235$$

Steps for Completing the Square

Use the distributive property to multiply x-10 by 920-2x and combine like terms.

$$3100=940x-2x^{2}-9200$$

Swap sides so that all variable terms are on the left hand side.

$$940x-2x^{2}-9200=3100$$

Add 9200 to both sides.

$$940x-2x^{2}=3100+9200$$

Add 3100 and 9200 to get 12300.

$$940x-2x^{2}=12300$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-2x^{2}+940x=12300$$

Divide both sides by -2.

$$\frac{-2x^{2}+940x}{-2}=\frac{12300}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{940}{-2}x=\frac{12300}{-2}$$

Divide 940 by -2.

$$x^{2}-470x=\frac{12300}{-2}$$

Divide 12300 by -2.

$$x^{2}-470x=-6150$$

Divide -470, the coefficient of the x term, by 2 to get -235. Then add the square of -235 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-470x+\left(-235\right)^{2}=-6150+\left(-235\right)^{2}$$

Square -235.

$$x^{2}-470x+55225=-6150+55225$$

Add -6150 to 55225.

$$x^{2}-470x+55225=49075$$

Factor x^{2}-470x+55225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-235\right)^{2}=49075$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-235\right)^{2}}=\sqrt{49075}$$

Simplify.

$$x-235=5\sqrt{1963} x-235=-5\sqrt{1963}$$

Add 235 to both sides of the equation.

$$x=5\sqrt{1963}+235 x=235-5\sqrt{1963}$$